Is 14977:2001 *II* mG-RTFl-awTwmlRd fa-%QTw-R-m Indian Standard CONTROL CHARTS BASED ON INSPECTION BY GAUGING ICS 03.120.30 0 BIS 2001 BUREAU OF INDIAN STANDARDS SHAH ZAFAR MARG MANAK BHAVAN, 9 BAHADUR NEW DELHI 110002 October 2001 Price Group 4 Statistical Methods for Quality and Reliability Sectional Committee, MSD 3 FOREWORD This Indian Standard was adopted by the Bureau of Indian Standards, after the draft finalized by the Statistical Methods for Quality and Reliability Sectional Committee had been approved by the Management and Systems Division Council. Control charts based on inspection by gauging can be conveniently used for controlling both the location and the variability parameters of a measurable characteristic of a process, having two-sided specifications. These control charts are used in situations where: a) b) c) d) the underlying distribution is normal or nearly normal, the location and the variability parameters are two-sided and one-sided (larger than the aimed value shifls) respectively, the values of the process mean and process variation are known, and inspection by attributes is preferred to that by variable from practical considerations. These charts are particularly suitable when underlying inspection is destructive in nature. The composition of the Committee responsible for formulation of this standard is given in Annex A. In reporting the result of a test or analysis, if the final value, observed or calculated is to be rounded off, it shall be done in accordance with IS 2:1960 `~les for rounding off numerical values ( revised )'. IS 14977:2001 Indian Standard CONTROL CHARTS BASED ON INSPECTION BY GAUGING 4.1.2 Since UGL and LGL are usually different from Upper Specification Limit ( USL ) and Lower Specification Limit ( LSL ) respectively, it is to be noted that the items falling under categories 4.1 (a) and 4.1 (c) may not necessarily be non-conforming. 1 SCOPE This standard covers control charts when inspection is based on gauging. The use of this control chart has been illustrated with an example. Further the advantages of this chart over X­ R charts and also the estimation of population mean and population variation have been included. 2 REFERENCES The following standards contain provisions, which through reference in this text constitute provisions of this standard. At the time of publication, the editions indicated were valid. All standards are subject to revision and parties to agreements based on this standard are encouraged to investigate the possibility of applying the most recent editions of the standards indicated below: 1SNo. 397 ( Part 1 ) : 1972 7920 Title 4.2 Besides mechanical inspection of engineering items for dimensional requirements by using pair of go-nogo gauges, there are many other situations in which inspection results generate this t~pe of data, for example, classi&ing the items by weight into underweight, normal, and over-weight; classi~ing fiseheads by their sensitivity into insensitive, normal and hypersensitive, etc. 4.3 Gauge-Limits 4.3.1 When the measurable quality characteristic follows normal or any other symmetrical distribution, LGL and UGL are placed symmetrically around the targeted process average. Thus if POand 00 are the known values for process mean and process standard deviation respectively, the gauge-limits are taken as: LGL = PO­ vao UGL = PO+ vao 4.3.2 The choice of v, when the underlying distribution is normal, has been discussed in 6. 4.4 For symmetrical distribution such as normal, when a sample of size n is gauged against a pair of gauges of the type mentioned in 4.3.1, the numbers a, ( below LGL ) and b ( above UGL ) are obtained. The measures ( b­ a) and ( b + a) are sensitive to changes in process mean ( p ) and process standard deviation ( o ) respectively from their corresponding target values. 5 PRELIMINARY STEPS 5.1 Choice of Probability of False Alarm Method for statistical quality control during production: Part 1 Control charts for variables (first revision ) Statistical vocabulary and Symbols: (Part 1): 1994 Probability and general statistical terms ( second revision) (Part2 ):1994 Statistical quality control (second revision 3 TERMINOLOGY ) For the purpose of this standard, the definitions given in IS 7920 ( Part 1 ) and IS 7920 ( Part 2 ) shall apply. 4 INSPECTION BY GAUGING 4.1 Inspection by gauging classifies items by using a pair of gauges, called Lower Gauge Limit ( LGL ) and Upper Gauge Limit ( UGL ), into the following three categories: a) b) c) those falling short of LGL, those lying between LGL and UGL, and those exceeding the UGL. 4.1.1 The number of items in a sample of size n belonging to each of the above three `categories are denoted by a, c, and b respectively. Of these, numbers a and b are directly noted, and c is taken as n-a-b. 1 Control limits for control charts based on inspection by gauging are set up ensuring a pre-assigned probability ct of falsely rejecting state of control. Here the concept of 3 c control limits is not pertinent. As such, a decision on the choice of a suitable in a given situation is to be taken. While a = 0.005 and 0.01 is normally suitable for the control chart purpose, a =0.01, 0.025 and 0.05 may also be considered if the tool is used for hypothesis testing. 5.2 Choice of Sub-group Sue Control charts based on inspection by gauging are IS 14977:2001 essentially attribute charts, similar to the conventional np charts. Thus, usually, a higher sub-group size, compared to control charts for variables, is needed to ensure reasonable protection against wrong decisions. However, through optimal determination of gauge-limits and control limits ( using criteria for decision under risk and/or uncertainty ), the errorcontrolling properties of these charts may compare quite favorably with those of variable control charts even for identical sample size. In view of this, and also to keep sub-group size small, often a sample size between 7 to 10 should be adequate, contrary to the higher sample size for an attribute control chart. 5.3 Choice of Control Charts 5.3.1 As mentioned in 4.4, for a sample gauged against a pair of gauges, ( b-a) and ( b + a ) are sensitive to changes in p and o. Hence a pair of control charts for ( b ­ a ) and ( b + a ) should be looked upon as substitutes for the conventional ~and R ( ors ) charts respectively. where POand 00 are the known values of process mean and process standard deviation. 6.2 A control chart is drawn with the x-axis meant for sub-group numbers and the y-axis meant for both a and b. 6.2.1 A single control limit is drawn at y = r. There is no central line. 6.3 Plotting Items in each sub-group are gauged one by one against LGL ( PO ­ v ao) and UGL ( UO + v GO ) sequentially, and the numbers a ( items below LGL ) and b ( items exceeding UGL ) are noted. Against each sub-group number, two points A and 9 are plotted corresponding Successive points to a and b respectively. corresponding to each of a and b maybe connected by continuous and broken lines separately, preferably using two different colours. Table 1 Values of Factors v and r for Different (n, a) Combinations (Clause.s6.l.1, 6.1.2 and6.1.3 a = 0.005 5.3.2 However, it has been found that a single control chart for a and b ( plotting two points corresponding to these measures against each sub-group number) is not only more user-tliendly but also has better errorcontrolling properties as compared to a pair of charts for ( b - a ) and ( b + a ), almost everywhere in the two-dimensional parameter space for y and cr,except on the line c = cro( that is, when o does not change ). In this case the latter pair of charts performs only marginally better. 5.3.3 Thus, it is recommended that a single chart for ( a,b ) should be used. 6 (a, h ) CHART ) a = 0.05 v 2.236 2.388 1.493 1.618 1.713 1.789 1.852 1.440 1.500 1.712 1.850 1.950 1.779 1.579 1.367 r 1 1 2 2 2 2 2 3 3 3 3 3 4 6 9 a = 0.010 v 2.806 1.735 1.889 1.997 2.081 1.595 1.670 1.732 1.786 1.657 1.802 1.907 1.989 1.611 1.507 r 1 2 2 2 2 3 3 3 3 4 4 4 4 7 9 a = 0.025 v 2,495 1.506 1.673 1.790 1.879 1.951 1.506 1.573 1.630 1.833 1.965 1.690 1.779 1.579 1.367 r 1 2 2 2 2 2 3 3 3 3 3 4 4 6 9 nv 2 3.023 3 1.894 r 1 2 2 2 3 3 3 3 3 4 4 5 6 8 11 4 2.040 5 2.113 6 7 8 1.626 1.713 1.784 1.844 1.896 1.747 1.889 1.762 1.672 1.547 1.379 6.1 Setting 9 10 15 20 25 30 40 50 Having decided on the sub-group size n and the probability ct of false alarm, find the values of the gauge factor v and the single control limit r from Table 1. 6.1.1 6.1.2 The values of v and r for various combinations of ( n, a ) have been given in Table 1 under the assumption that the distribution of quality characteristic is normal. 6.1.3 Since usually the optimal v values are smaller than 3, as is evident from Table 1, the gauges are somewhat compressed or narrow as compared to standard tolerances. Therefore, these charts are sometimes referred to as compressed or narrow limit gauging charts. 6.1.4 Gauge-limits are set as: LGL=pO­vaO and UGL=pO+vcO 2 6.4 Interpretation 6.4.1 The status of a sub-group will be indicated in the control chart as follows: S 1: both a and b lie below the control limit; S2: a lies below the control lirni~ b lies on or above control limit; IS 14977:2001 S3: b lies below the control limit, a lies on or above control limit; and S4 : both a and b lie on or above control limit. S 1 indicates a state of control, while the other three indicate presence of assignable causes of variation in the process. S2 indicates a shift of process mean ( P ) to the right of u~. S3 indicates a shifi of process mean to the left of PO,and S4 indicates a sKIR( increase ) in process variation from (crO ). When the process is in a state of control, the two lines connecting a and b values are expected to frequently intersect each other. If they remain separated for a considerable period, even below the control limit, an ensuing shitl in location may be suspected. 6.4.2 7 ESTIMATION OF PROCESS STANDARD DEVIATION MEAN AND actual measurements were not needed. However, to facilitate comparison, these results have been retained. 8.3 For installing an ( a, b ) chart, the value a = 0.005 may be taken as it is nearest to the corresponding value of u in control charts for variables using 3c$Iinits. Forcx=0.005, n=7, from Table l,v= 1.713 andr=3. Also PO = 19.5 andoo= 1.00 Hence LGL= 19.5 ­ 1.713 = 17.787 UGL=19.5 +1.713=21.213 8.4 For each sample, the items are gauged sequentially against LGL and UGL and the values of a and b are noted. These values are shown in CO19 and 10 of Table 2. 8.5 The resultant control chart is shown in Fig. 1. 8.6 It appears from the chart that the points corresponding to b for control units number 14 and 15were found to have gone out of control limit. Suitable remedial action was taken to stabilize the process in terms of location after which the process was again found to be in a state of statistical control. 8.7 Comparison with Y-R Chart 7.1 Collect a and b values from k sub-groups taken from a process under state of statistical control, where n is the constant sub-group size, and t( pz ), t (1 ­ ~ ) are the values of standard normal variates corresponding to areas to the letl of pr and 1 ­ ~ respectively, to be obtained from standard normal tables. t(~) will be negative and t(1­pr) will be positive. 7.2 The estimates of process mean(p) and process standard deviation (o) are obtained as: ~= L t(l-p~)uf(P~) 8.7.1 If (~-R) chart had been applied for data given in Table 2, the control limits for X-chart and R-chart are given below: X- chart R - chart ucL=po+Acro=19.5+ 1.134=20.634 LcL=po­AcJo=19.5+ 1.134=18.366 UCL = Dz a. = 5.203 LCL = DI 00 = 0.205 t(l­~)and o = U-L t(fi) t(l­~)­ t(pz) 8.7.2 For values of A, D, and D2, reference maybe made to IS 397 (Part 1 ). 8.7.3 From the ~and R values given in the last two columns of Table 2, it is clear, that in ~-chart also, the points going out of control are only those corresponding to sub-group numbers 14 and 15. 8.7.4 Just as no sub-group in ( a, b ) chart shows that both a and b values are out of control, no point in R-chart indicates an out of control situation for ,, process variation. 8.8 Similarly it can be shown from median chart that the sub-group numbers 14 and 15 only would go out of control. 8.9 Thus, for this example (a, b ) chart is found to be as sensitive as either an (~­ R) chart or a ( Median, range ) chart. 3 where U and L are the UGL and LGL respectively. 8 EXAMPLE FOR (a, b ) CHART 8.1 In a certain firm producing composite conductors of size 7/2.79 mm, it was decided to install suitable control charts for ultimate tensile strength expressed in units of kgf/mm2. From past data, the process average was to be centered around at 19.5 kgf7mm2. It was also known that the standard deviation of the process was 1.00 kgf/mm2. From each cable drum chosen as the control unit, 7 sample results were available corresponding to the 7 strands of the composite conductor. 8.2 The results are given in COI(2) to (8) of Table 2. It maybe noted that for operation of ( a, b ) chart, the IS 14977:2001 Table 2 Control Chart Data on Ultimate Tensile Strength ( kgf/mm2 ) of Composite Conductor Size 7/2.79 mm ( Clauses 8.2,8 .4,8.7.1 Cable and8.7.3 ) Mean x Range R of the Item Number in Sample 0 1 2 (3) 19.30 19.14 18.98 18.81 18.80 18.00 18.65 18.32 18.32 18.00 20.92 18.00 18.61 22.90 19.47 18.32 19.14 18.80 17.79 3 (4) 18.81 19.96 18.32 18.65 18.81 18.00 18.00 19.63 20.78 21.74 19.96 17.34 18.28 20.94 22.90 19.63 18.32 21.76 20.42 4 (5) 18.98 18.00 19.63 18.98 18.98 18.32 18.32 19.80 19.63 19.27 19.96 18.65 18.61 21.60 23.39 18.12 19.60 19.27 18.32 5 (6) 19.30 18.49 19.30 18.81 18.32 19.96 18.49 18.00 19.43 18.65 20.78 19.14 18.32 19.96 18.32 18.81 17.79 21.25 19.76 6 (7) 18.98 18.32 19.30 18.81 18.02 19.30 17.51 22.74 19.63 20.75 21.08 18.32 18.32 21.27 22.90 19.63 19.30 20.94 17.29 7 (8) 19.80 19.80 19.30 19.47 18.81 19.80 18.65 19.14 20.94 19.96 20.78 18.00' 17.62 21.90 22.90 20.12 18.28 20.29 18.28 (9) o 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 2 (lo) o 0 0 0 0 0 0 1 0 1 0 0 0 5 5 0 0 2 0 b Drum No. (1) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. (2) 19.63 19.47 19.14 18.49 18.32 18.81 19.14 18.98 19.80 19.11 18.32 18.16 18.12 22.90 22.09 18.00 17.83 19.96 16.80 (11) 19.26 19.03 19.14 18.86 18.58 18.88 18.39 19.52 19.79 19.64 20.26 18.23 18.27 21.64 21.71 18.95 18.61 20.47 18.38 (12) 0.99 1.96 1.31 0.98 0.96 1.96 1.63 4.74 2.62 3.74 2.76 1.80 0.99 2.94 5.07 2.12 1.81 2.49 3.62 1234567 091011121314 1516171319 DRUMNo, *a FIG. 1 (a, b ) CHARTS +b 4 IS 14977:2001 9 ESTIMATION OF PROCESS MEAN AND PROCESS STANDARD DEVIATION a) b) c) accept HO, if Ci= n-r+ 1 reject HO: if ai or bi = r, or continue gauging an additional item. As the sub-group numbers 14 and 15 are not in statistical control, these sub-groups are eliminated while estimating the process mean p and process standard deviation o from the (a, b ) chart: ;=6/17 =0.353 ~=4117=0.235 so that &=0.35317 =.0504 l­~=0.9664 and t (p=) =­1.640 78, ~= 0.235/7= 0.336 t(/­p5)=l.42141 10.5 If HOis neither accepted nor rejected during the gauging of first n- 1 items, the nth item is finally gauged, and one of the following two decisions is taken: a) b) reject HO,if an or bn = r, and accept HO,otherwise. Hence using the formula given in 4.5.2, we get p= 21 .213x 1.64078+ 17.787x 1.42141 1.42141+1.64078 and CT= 21.213­17.787 1.421 41+ 1.64078 = 1.119 TESTING TOOL = 19.623 10.6 Thus at most n items for each sub-group of size and the average amount of inspection will be smaller than n. The amount of saving depends upon n, cxand the amount of shifts in v and cr. This feature is particularly appealing when inspection is destructive in nature and/or time consuming, and observations in the sample are available in natural sequence, for example, in life testing. n are to be gauged, 11 COMPARISON WITH (~-11) OR(~­s) CHART 11.1 Although ( a, b ) chart is essentially a control chart for attributes, because of the optimal determination of gauge-limits using criteria for decisionmaking under risk and/or uncertainty, the error controlling properties of the chart have been found to compare quite favorably with those of ( ~­ R ) and ( ~­s ) charts. For small sub-group sizes, (a, b) chart with sub-group size n + 1 will match the performance of an ( ~­ R )-chart or ( ~­s ) chart with sub-group size n, largely belying the popular belief to the contrary. 11.2 The chart enjoys a distinct edge over those based on ( ~­ R ) or ( ~­s ) on the two counts. First, here inspection, record keeping and analysis of results are much simple and less time-consuming, and therefore more economical and user-friendly. Secondly, unlike the latter two, this is adaptable to curtailed inverse sampling scheme, resulting in salvaging of substantial inspection material and saving of time, when inspection . is destructive in nature and/or time-consuming. 10 USE AS A HYPOTHESIS 10.1 The ( a, b ) chart outlined above may also be used as a tool for testing HO: L = po, c < croagainst HI : II # PO,0>00 under a normal model. The null hypothesis is to be rejected if either a 2 r or b 2 r or both. 10.2 However, in suth a situation, it may not be necessary to gauge the entire sample if a curtailed inverse sampling scheme, as discussed below is adhered to. 10.3 Here items are to be gauged one by one, at the ith stage note the values of ai and bi, the numbers of observations upto the ith item falling below LGL and exceeding UGL respectively, and Ci = i ­ ai ­ bi, i=l,2 .. ..n-l. 10.4 At the ith stage, one of the following 3 decisions is taken: IS 14977:2001 ANNEX A ( Foreword) COMMITTEE COMPOSITION Statistical Methods for Quality and Reliability Sectional Committee, MSD 3 Organization Representative(s) PROF S. P. MUKHERJEE ( SHRI B. N. JHA SHRI R. S. BHARGAVA SHRIA. K. SRIVASTAVA ( Alternate SHRI S. N. JHA SHRIA. V. KRISHNAN( CalcuttaUniversity,Kolkata AseaBrownBoveri Limited,Bangalore BajajAuto Limited,Pune Bharat HeavyElectrical Limited,Hyderabad Continental Device India Limited, New Delhi Directorate General of Quality Assurance, New Delhi Chairman ) ) Alternate ) Alternate ) SHRI G. V. SUBRAMANIAN SHRIMATI RENU NEHRU( SHRI S. K. SRIVASTAVA LT COL P. VIJAYAN ( DR ASHOKKUMAR SHRI C. S. V. NARENDRA Ms N. V. NAIK Alternate ) Directorate of Standardization, Ministry of Defence, New Delhi Escorts Limited, Faridabad HMT Limited, R & D Centre, Bangalore Indian Agricultural Statistics Research Institute, New Delhi DR S. D. SHARMA DR A. K. SRIVASTAVA ( Alternate DR B. DAS DR DEBABRATA RAY ( ) Indian Association for Productivity, ( IAPQR), Kolkata Indian Institute of Management, Quality and Reliability Alternate) Lucknow Kolkata PROFS. C. CHAJCRABORTY SHRI U. DUTTA DR S. N. PAL ( Indian Jute Industries' Research Association, Alternate ) Alternate) Alternate ) Indian Statistical Institute, Kolkata PROFS. R. MOHAN PROFARVINDSETH ( Lucas-TVS Limited, Chennai SHRI N. S. SREENIVASAN SHRI G. VIJAYAKUMAR( SHRIY. K. BHAT SHRI G. W. DATEY ( National Institution for Quality and Reliability, New Delhi Alternate ) ) Powergrid Corporation of India Limited, New Delhi . DR S. K. AGARWAL SHRI D. CHAKRABORTY ( Alternate SHRI A. SANJEEVA RAO SHRI RAMANI SUBRAMANIAN ( SRF Limited, Chennai Alternate ) Standardization, New Delhi Tata Engineering Jamshedpur Testing and Quality Certification Directorate, SHRI S. K. KIMOTHI SHRI P. N. SRIKANTH ( SHRI S. N. DAS SHRI SHANTK SARUP ( Alternate ) and Locomotive Company Limited ( TELCO ), Alternate ) ( Continued on page 7 ] 6 IS 14977:2001 ( Continuedfrom page 6 ) Organization University of Delhi, Delhi In personal capacity ( B -109 Malviya Nagac New Delhi -110 017) Impersonal capacity (20/I New Delhi -110029 ) Directorate General BIS Representative(s) PROFM. C. AGRAWAL PROFA. N. NANKANA SHRI D. R. SEN Krishna Nagac Safdarjung Enclave SHRTP. K. GAMBHtR, Director and Head & Member Secretary ~Representing Director General, BIS ( Ex-officio Member ) ] Basic Statistical Methods Subcommittee ( MSD 3:1 ) Calcutta University, Kolkata Bajaj Auto Limited, Pune Directorate of Standardization, Ministry of Defence, New Delhi Quality and Reliability PROFS. P. MUKHERJEE ( Convener ) SHRI A. K. SRWASTAVA DR ASHOIC KUMAR Indian Association for Productivity, (IAPQR), Kolkata Indian Statistical Institute, Kolkata DR B. DAS DR A. LAHIRI( Alternate ) PROFS. R. MOHAN SHRI Y. K. BHAT SHRI G. W. DATEY ( Alternate ) DR S. K. AGARWAL National Institution for Quality and Reliability, New Delhi Powergrid Corporation Standardization, New Delhi Tata Engineering Jamshedpur of India Limited, New Delhi Directorate, Testing and Quality Certification SHRIS. K. KIMOTHI and Locomotive Company Limited (TELCO), SHRI SHANT1SARtJP University College of Medical Sciences (UCMS), Delhi University of Delhi, Delhi In personal capacity ( B -109 Malviya Naga~ New Delhi -110 017) In personal capacity ( 20/1 Krishna Nagar Safdarjung Enclave, ) New Delhi -110029 DR A. INDRAYAN PROFM. C. AGRAWAL PROFA. N. NANKANA SHRt D. R. SEN Bureau of Indian Standards BIS is a statutory institution established under the Bureau of Indian Standards Act, 1986 to promote harmonious development of the activities of standardization, marking and quality certification of goods and attending to connected matters in the country. Copyright BIS has the copyright of all its publications. No part of these publications maybe reproduced in any form without the prior permission in writing of BIS. This does not preclude the free use, in the course of implementing the standard, of necessary details, such as symbols and sizes, type or grade designations. Enquiries relating to copyright be addressed to the Director (Publications), BIS. Review of Indian Standards Amendments are issued to standards as the need arises on the basis of comments. Standards are also reviewed periodically; a standard along with amendments is reaffirmed when such review indicates that no changes are needed; if the review indicates that changes are needed, it is taken un for revision. Users of Indian Standards should ascertain that they are in possession of the latest amendme. r edition by referring to the latest issue of `BIS Catalogue' and `Standards : Monthly Additions'. This Indian Standard has been developed from Doc : No. MSD 3 ( 150 ). Amendments Issued Since Publication Amend No. Date of Issue Text Affected BUREAU OF INDIAN STANDARDS Headquarters: Manak Bhavan, 9 Bahadur Shah Zafar Marg, New Delhi 110002 Telephones: 3230131,3233375,3239402 Regional Offices: Central: Manak Bhavan, 9 Bahadur Shah Zafar Marg NEW DELHI 110002 Eastern: 1/14 C. I. T. Scheme VII M, V. 1.P. Road, Kankurgachi KOLKATA 700054 Telegrams: Manaksanstha ( Common to all offices) Telephone 3237617 { 3233841 3378499,3378561 { 3378626,3379120 603843 { 602025 2541216,2541442 { 2542519,2541315 8329295,8327858 { 8327891,8327892 Northern: SCO 335-336, Sector 34-A, CHANDIGARH 160022 Southern: C. 1.T. Campus, IV Cross Road, CHENNAI 600113 Western : Manakalaya, E9 MIDC, Marol, Andheri (East) MUMBA1400 093 Branches : AHMADABAD. BANGALORE. BHOPAL. BHUBANESHWAR. COIMBATORE. FARIDABAD. GHAZIABAD. GUWAHATI. HYDERABAD. JAIPUR. KANPUR. LUCKNOW. NAGPUR. NALAGARH. PATNA. PUNE. RAJKOT. THIRUVANANTHAPURAM. Printed at New India Printing Press, Khurja, India